<<

. 29
( 42 .)



>>


The energy spectrum corresponding to the Coulomb potential
V (r) = ’ 1 is
r
1 1
ElN = ’ (6.69)
2 N + l + D’1 2
2

In (6.69) N and l stand for the radial and angular momentum quan-
tum numbers respectively.
From (6.30) we obtain
1 1
G=’ 2
2 N +l+ D’1
2
1 1
+ 2
2 l + D’1
2
1 1
+ (6.70)
2
2 N ’1+l+j+ D’1
2



© 2001 by Chapman & Hall/CRC
Setting j = 1 it is easy to realize that G becomes independent
of N leading to
1 1
G= (6.71)
2 l + D’1 2
2

Then the general results for H± in the D dimensional space are

1 d2 ±l
El0
H+ ≡ Hl ’ =’ +2
2 dr2 2r
’2
11 1
’+ l + (D ’ 1) (6.72)
r2 2
1 d2 ±l+1
H’ ≡ Hl+1 + G = ’ +
2 dr2 2r2
’2
11 1
’+ l + (D ’ 1) (6.73)
r2 2
where ±l is given by (6.68). For D = 3 these are in agreement with
(6.38) and (6.39).

(b) Isotropic oscillator potential

For the isotropic oscillator potential V (r) = 1 r2 the energy levels
2
are
1
ElN = 2N + l + D , D ≥ 2 (6.74)
2
From (6.30) we ¬nd
D
G=2’ l+j+ (6.75)
2
which turns out to be independent of N . It gives the following
isospectral Hamiltonians

1 d2 ±l
H+ =’ +2
2 dr2 2r
1 D
+ r2 ’ l + (6.76)
2 2
1 d2 ±l+j
H’ =’ +
2 dr2 2r2
1 D
+ r2 ’ l + j + +2 (6.77)
2 2


© 2001 by Chapman & Hall/CRC
These may be compared with (6.43) and (6.44) for D = 3.
To conclude it is worthwhile to note that transformations from
the Coulomb problem to the isotropic oscillator and vice-versa can
be carried out [13-23] and the results turn out to be generalizable to
D-dimensions [13-24].


6.5 References
[1] B.H. Bransden and C.J. Joachain, Introduction to Quantum
Mechanics, ULBS, Longman Group, Essex, UK, 1984.

[2] V.A. Kostelecky and M.M. Nieto, Phys. Rev. Lett., 53, 2285,
1984.

[3] V.A. Kostelecky and M.M. Nieto, Phys. Rev. Lett., 56, 96,
1986.

[4] V.A. Kostelecky and M.M. Nieto, Phys. Rev., A32, 1293, 1985.

[5] R.W. Haymaker and A.R.P. Rau, Am. J. Phys., 54, 928, 1986.

[6] A.R.P. Rau, Phys. Rev. Lett., 56, 95, 1986.

[7] A. Lahiri, P.K. Roy, and B. Bagchi, J. Phys. A. Math. Gen.,
20, 3825, 1987.

[8] J.D. Newmarch and R.H. Golding, Am. J. Phys., 46, 658,
1978.

[9] A. Lahiri, P.K. Roy, and B. Bagchi, Phys. Rev., A38, 6419,
1988.

[10] A. Lahiri, P.K. Roy, and B. Bagchi, Int. J. Theor. Phys., 28,
183, 1989.

[11] A.B. Balantekin, Ann. Phys., 164, 277, 1985.

[12] H. Ui and G. Takeda, Prog. Theor. Phys., 72, 266, 1984.

[13] V.A. Kostelecky, M.M. Nieto, and D.R. Truax, Phys. Rev.,
D32, 2627, 1985.


© 2001 by Chapman & Hall/CRC
[14] B. Baumgartner, H. Grosse, and A. Martin, Nucl. Phys., B254,
528, 1985.

[15] A. Lahiri, P.K. Roy, and B. Bagchi, J. Phys. A: Math. Gen.,
20, 5403, 1987.

[16] J.D. Louck and W.H. Scha¬er, J. Mol. Spectrosc, 4, 285, 1960.

[17] J.D. Louck, J. Mol. Spectrosc, 4, 298, 1960.

[18] J.D. Louck, J. Mol. Spectrosc, 4, 334, 1960.

[19] D. Bergmann and Y. Frishman, J. Math. Phys., 6, 1855, 1965.

[20] D.S. Bateman, C. Boyd, and B. Dutta-Roy, Am. J. Phys., 60,
833, 1992.

[21] P. Pradhan, Am. J. Phys., 63, 664, 1995.

[22] H.A. Mavromatis, Am. J. Phys., 64, 1074, 1996.

[23] A. De, Study of a Class of Potential in Quantum Mechanics,
Dissertation, University of Calcutta, Calcutta, 1997.

[24] A. Chatterjee, Phys. Rep., 186, 249, 1990.




© 2001 by Chapman & Hall/CRC
CHAPTER 7

Supersymmetry in
Nonlinear Systems

7.1 The KdV Equation
One of the oldest known evolution equations is the KdV, named after
its discoverers Korteweg and de Vries [1], which governs the motion
of weakly nonlinear long waves. If δ(x, t) is the elevation of the
water surface above some equilibrium level h and ± is a parameter
characterizing the motion of the medium, then the dynamics of the
¬‚ow can be described by an equation of the form

3 g 2 1
δt = δδx + ±δx + σδxxx (7.1)
2 h 3 3
where the su¬xes denote partial derivatives with respect to the space
(x) and time (t) variables. The parameter σ signi¬es the relationship
3
between the surface tension T of the ¬‚uid and its density ρ : h ’ hT .
3 ρg
It can be easily seen that (7.1) can be put in a more elegant form

ut = 6uux ’ uxxx (7.2)

by a simple transformation of the variable δ and scaling the param-
eters h, ±, and σ appropriately. In the literature (7.2) is customarily
referred to as the KdV equation. Some typical features which (7.2)
exhibit are


© 2001 by Chapman & Hall/CRC
(i) Galilean invariance: The transformations u (x , t ) ’
u(x, t)+ u0 where x ’ x ± u0 t and t ’ t leave the form of (7.2)
6
unchanged.

(ii) PT symmetry: Both u(x, t) and u(’x, ’t) are solutions of
(7.2).

(iii) Solitonic solution: Equation (7.2) possesses a solitary
wave solution

a a
u(x, t) = ’ sech2 (x ’ at) , a ∈ R (7.3)
2 2

which happens to be a solitonic solution as well.
Solitary waves in general occur due to a subtle interplay between
the steepening of nonlinear waves and linear dispersive e¬ects [2].
Sometimes solitary waves are also form-preserving, these are then
referred to as solitons. Solitons re¬‚ect particle-like behaviour in that
they proceed almost freely, can collide among themselves very much
like the particles do, and maintain their original shapes and velocities
even after mutual interactions are over.
It was Scott-Russell who ¬rst noted a solitary wave while observ-
ing the motion of a boat. In a classic paper [3], Scott-Russell gives a
fascinating account of his chance meeting with the solitary wave in
the following words

. . . I was observing the motion of a boat which was rapidly drawn
along a narrow channel by a pair of horses, when the boat suddenly
stopped (but) not so the mass of the water in the channel which it had
put in motion; it accumulated around the prow of the vessel in a state
of violent agitation, then suddenly leaving it behind, rolled forward
with great velocity, assuming the form of a large solitary elevation,
a rounded, smooth and well-de¬ned heap of water, which continued
its course along the channel apparently without change of form or
diminution of speed.

We now know that what Scott-Russell saw was actually a solitary
wave. We also believe that the KdV equation provides an analytical
basis to his observations.


© 2001 by Chapman & Hall/CRC
Intense research on the KdV equation began soon after Gardner
and Morikawa [4] found an application in the problem of collision-free
hydromagnetic waves. Subsequently works on modelling of longitu-
dinal waves in one-dimensional lattice of equal mass were reported
[5,6] and numerical computations were carried out [7] to compare
with the recurrences observed in the Fermi-Pasta-Ulam model [8].
The relevance of KdV to describe pressure waves in a liquid gas
bubble chamber was also pointed out [9]. Moreover, the KdV was
found to play an important role in explaining the motion of the
three-dimensional water wave problem [10]. A number of theoretical
achievements were made alongside these. Gardner, Greene, Kruskal,
and Miura [11,12] developed a method of ¬nding an exact solution for
the initial-value problem. Lax [13] proposed an operator approach
towards interpreting nonlinear evolution equations in terms of con-
served quantities, and Zakharov and Shabat [14] formulated what is
called the inverse scattering approach. Further, it was also discov-
ered [15,16] that algebraic connections could be set up by means of
Backlund transformations implying that the solutions of certain evo-
lutionary equations are correlated. We do not intend to go into the
details of the various research directions which took o¬ from these
works but we will focus primarily on the symmetry principles like
conservation laws as well as the solutions of a few nonlinear equa-
tions which can now be analyzed in terms of the concepts of SUSY.
All this will form the subject matter of this chapter.


7.2 Conservation Laws in Nonlinear Systems
The existence of conservation laws leads to integrals of motion. A
conservation law is an equation of the type

Tt + χx = 0 (7.4)

where T is the conserved density and ’χ represents the ¬‚ux of the
¬‚ow. The KdV equation is expressible in the form (7.4), a property it

<<

. 29
( 42 .)



>>